Approximate calculation of frequencies of localized vibrations |
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Authors: | O Litzman J Celý |
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Institution: | (1) Institute of Theoretical Physics and Dept. of Solid State Physics, Faculty of Natural Sciences, Brno University, USSR |
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Abstract: | An exact calculation of the frequencies of localized vibrations in a crystal with defects meets with considerable difficulties and therefore only the simplest cases have been calculated 9, 10]. An approximate calculation is confined to computing the frequencies of a system composed of several atoms in the neighbourhood of the defect. The interaction between this system and the other atoms of the lattice, which however are immobile in this approximation, is taken into consideration. Since the analytical estimate of the approximation is very rough, the approximate and exact calculations have been compared for some concrete cases (Fig. 1, 2). It is clear from these calculations that the use of the proposed approximation leads to good results.
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![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![dcy](/content/w407l3044715g314/xxlarge1076.gif) ![pcy](/content/w407l3044715g314/xxlarge1087.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![lcy](/content/w407l3044715g314/xxlarge1083.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![gcy](/content/w407l3044715g314/xxlarge1075.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![yucy](/content/w407l3044715g314/xxlarge1102.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![scy](/content/w407l3044715g314/xxlarge1089.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![pcy](/content/w407l3044715g314/xxlarge1087.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![dcy](/content/w407l3044715g314/xxlarge1076.gif) ![vcy](/content/w407l3044715g314/xxlarge1074.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![zhcy](/content/w407l3044715g314/xxlarge1078.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![ycy](/content/w407l3044715g314/xxlarge1099.gif) ![mcy](/content/w407l3044715g314/xxlarge1084.gif) . ![Pcy](/content/w407l3044715g314/xxlarge1055.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![scy](/content/w407l3044715g314/xxlarge1089.gif) ![kcy](/content/w407l3044715g314/xxlarge1082.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![lcy](/content/w407l3044715g314/xxlarge1083.gif) ![softcy](/content/w407l3044715g314/xxlarge1100.gif) ![kcy](/content/w407l3044715g314/xxlarge1082.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![lcy](/content/w407l3044715g314/xxlarge1083.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![chcy](/content/w407l3044715g314/xxlarge1095.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![scy](/content/w407l3044715g314/xxlarge1089.gif) ![kcy](/content/w407l3044715g314/xxlarge1082.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![tscy](/content/w407l3044715g314/xxlarge1094.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![kcy](/content/w407l3044715g314/xxlarge1082.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![pcy](/content/w407l3044715g314/xxlarge1087.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![kcy](/content/w407l3044715g314/xxlarge1082.gif) ![scy](/content/w407l3044715g314/xxlarge1089.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![mcy](/content/w407l3044715g314/xxlarge1084.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![tscy](/content/w407l3044715g314/xxlarge1094.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![vcy](/content/w407l3044715g314/xxlarge1074.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![scy](/content/w407l3044715g314/xxlarge1089.gif) ![softcy](/content/w407l3044715g314/xxlarge1100.gif) ![mcy](/content/w407l3044715g314/xxlarge1084.gif) ![gcy](/content/w407l3044715g314/xxlarge1075.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![ucy](/content/w407l3044715g314/xxlarge1091.gif) ![bcy](/content/w407l3044715g314/xxlarge1073.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) , ![bcy](/content/w407l3044715g314/xxlarge1073.gif) ![ycy](/content/w407l3044715g314/xxlarge1099.gif) ![lcy](/content/w407l3044715g314/xxlarge1083.gif) ![pcy](/content/w407l3044715g314/xxlarge1087.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![vcy](/content/w407l3044715g314/xxlarge1074.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![dcy](/content/w407l3044715g314/xxlarge1076.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![scy](/content/w407l3044715g314/xxlarge1089.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![vcy](/content/w407l3044715g314/xxlarge1074.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![pcy](/content/w407l3044715g314/xxlarge1087.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![bcy](/content/w407l3044715g314/xxlarge1073.gif) ![lcy](/content/w407l3044715g314/xxlarge1083.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![zhcy](/content/w407l3044715g314/xxlarge1078.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![ycy](/content/w407l3044715g314/xxlarge1099.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![chcy](/content/w407l3044715g314/xxlarge1095.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![ycy](/content/w407l3044715g314/xxlarge1099.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![scy](/content/w407l3044715g314/xxlarge1089.gif) ![chcy](/content/w407l3044715g314/xxlarge1095.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![dcy](/content/w407l3044715g314/xxlarge1076.gif) ![lcy](/content/w407l3044715g314/xxlarge1083.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![kcy](/content/w407l3044715g314/xxlarge1082.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![ycy](/content/w407l3044715g314/xxlarge1099.gif) ![kcy](/content/w407l3044715g314/xxlarge1082.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![kcy](/content/w407l3044715g314/xxlarge1082.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![ycy](/content/w407l3044715g314/xxlarge1099.gif) ![scy](/content/w407l3044715g314/xxlarge1089.gif) ![lcy](/content/w407l3044715g314/xxlarge1083.gif) ![ucy](/content/w407l3044715g314/xxlarge1091.gif) ![chcy](/content/w407l3044715g314/xxlarge1095.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) (![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) . 1, 2). ![Icy](/content/w407l3044715g314/xxlarge1048.gif) ![ecy](/content/w407l3044715g314/xxlarge1101.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![scy](/content/w407l3044715g314/xxlarge1089.gif) ![chcy](/content/w407l3044715g314/xxlarge1095.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![vcy](/content/w407l3044715g314/xxlarge1074.gif) ![ycy](/content/w407l3044715g314/xxlarge1099.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![kcy](/content/w407l3044715g314/xxlarge1082.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) , ![chcy](/content/w407l3044715g314/xxlarge1095.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![pcy](/content/w407l3044715g314/xxlarge1087.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![mcy](/content/w407l3044715g314/xxlarge1084.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![pcy](/content/w407l3044715g314/xxlarge1087.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![bcy](/content/w407l3044715g314/xxlarge1073.gif) ![lcy](/content/w407l3044715g314/xxlarge1083.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![zhcy](/content/w407l3044715g314/xxlarge1078.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![ncy](/content/w407l3044715g314/xxlarge1085.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![gcy](/content/w407l3044715g314/xxlarge1075.gif) ![mcy](/content/w407l3044715g314/xxlarge1084.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![dcy](/content/w407l3044715g314/xxlarge1076.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![scy](/content/w407l3044715g314/xxlarge1089.gif) ![chcy](/content/w407l3044715g314/xxlarge1095.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![vcy](/content/w407l3044715g314/xxlarge1074.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![dcy](/content/w407l3044715g314/xxlarge1076.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![khcy](/content/w407l3044715g314/xxlarge1093.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![ocy](/content/w407l3044715g314/xxlarge1086.gif) ![shcy](/content/w407l3044715g314/xxlarge1096.gif) ![icy](/content/w407l3044715g314/xxlarge1080.gif) ![rcy](/content/w407l3044715g314/xxlarge1088.gif) ![iecy](/content/w407l3044715g314/xxlarge1077.gif) ![zcy](/content/w407l3044715g314/xxlarge1079.gif) ![ucy](/content/w407l3044715g314/xxlarge1091.gif) ![lcy](/content/w407l3044715g314/xxlarge1083.gif) ![softcy](/content/w407l3044715g314/xxlarge1100.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) ![tcy](/content/w407l3044715g314/xxlarge1090.gif) ![acy](/content/w407l3044715g314/xxlarge1072.gif) .
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